Introduction to Modern Economic Growth 7.9 References and Literature The main material covered in this chapter is the topic of many excellent applied mathematics and engineering books The purpose here has been to provide a review of the results that are most relevant for economists, together with simplified versions of the most important proofs The first part of the chapter is closer to the calculus of variations theory, because it makes use of variational arguments combined with continuity properties Nevertheless, most economists not need to study the calculus of variations in detail, both because it has been superseded by optimal control theory and also because most of the natural applications of the calculus of variations are in physics and other natural sciences The interested reader can look at Gelfand and Fomin (2000) Chiang (1992) provides a readable and simple introduction to the calculus of variations with economic examples The theory of optimal control was originally developed by Pontryagin et al (1962) For this reason, the main necessary condition is also referred to as the Pontryagin’s (Maximum) Principle The type of problem considered here (and in economics more generally) is referred to as the Lagrange problem in optimal control theory The Maximum Principle is generally stated either for the somewhat simpler Meyer problem or the more general Bolza problem, though all of these problems are essentially equivalent, and when the problem is formulated in vector form, one can easily go back and forth between these different problems by simple transformations There are several books with varying levels of difficulty dealing with optimal control Many of these books are not easy to read, but are also not entirely rigorous in their proofs An excellent source that provides an advanced and complete treatment is Fleming and Rishel (1975) The first part of this book provides a complete (but rather different) proof of Pontryagin’s Maximum Principle and various applications It also provides a number of theorems on existence and continuity of optimal controls A deeper understanding of sufficient conditions for existence of solution and the structure of necessary conditions can be gained from the excellent (but abstract and difficult) book by Luenberger (1969) The results in this book are general enough to cover both discrete time and continuous time dynamic optimization This book also gives a very good sense of why maximization in function spaces is 361